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Sliding-mode control of uncertain systems in the presence of unmatcheddisturbances with applicationsX. Hana; E. Fridmanb; S. K. Spurgeona

a Instrumentation, Control and Embedded Systems Research Group, School of Engineering and DigitalArts, Kent University, Canterbury, Kent, UK b School of Electrical Engineering, Tel Aviv University,Tel Aviv, Israel

First published on: 15 November 2010

To cite this Article Han, X. , Fridman, E. and Spurgeon, S. K.(2010) 'Sliding-mode control of uncertain systems in thepresence of unmatched disturbances with applications', International Journal of Control,, First published on: 15November 2010 (iFirst)To link to this Article: DOI: 10.1080/00207179.2010.527375URL: http://dx.doi.org/10.1080/00207179.2010.527375

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International Journal of Control2010, 1–14, iFirst

Sliding-mode control of uncertain systems in the presence of unmatched

disturbances with applications

X. Hana, E. Fridmanb and S.K. Spurgeona*

aInstrumentation, Control and Embedded Systems Research Group, School of Engineeringand Digital Arts, Kent University, Canterbury, Kent, UK; bSchool of Electrical Engineering,

Tel Aviv University, Tel Aviv, Israel

(Received 2 June 2010; final version received 23 September 2010)

This article considers the development of constructive sliding-mode control strategies based on measured outputinformation only for linear, time-delay systems with bounded disturbances that are not necessarily matched.The novel feature of the method is that linear matrix inequalities are derived to compute solutions to both theexistence problem and the finite time reachability problem that minimise the ultimate bound of the reduced-ordersliding-mode dynamics in the presence of state time-varying delay and unmatched disturbances. Themethodology provides guarantees on the level of closed-loop performance that will be achieved by uncertainsystems which experience delay. The methodology is also shown to facilitate sliding-mode controller design forsystems with polytopic uncertainties, where the uncertainty may appear in all blocks of the system matrices.A time-delay model with polytopic uncertainties from the literature provides a tutorial example of the proposedmethod. A case study involving the practical application of the design methodology in the area of autonomousvehicle control is also presented.

Keywords: sliding-mode control; output feedback; state delay; LMIs

1. Introduction

The control of time-delay systems is known to be ofpractical significance. Problems largely fall into twocategories. The first category arises because of the needto model systems more accurately given increasingperformance expectations. Many processes, such asmanufacturing processes, include such after effectphenomena in their dynamics and time delay is alsoproduced via the actuators, sensors and networksinvolved in the practical implementation of feedbackcontrol strategies. The second class of problems ariseswhen time delays are used as a modelling tool tosimplify some infinite-dimensional systems. Thisapproach is used for constructing models of distributedsystems modelled by partial differential equationswhere a set of finite-dimensional state variables withappropriate time-delay characteristics can be used torepresent heat exchange processes, for example.

The application of sliding-mode control (SMC) tothe problem of systems with time delay is a far fromtrivial problem generically, involving the combinationof delay phenomenon with relay actuators which hasthe potential to induce oscillations around the slidingsurface during the sliding-mode. There are a numberof papers which have considered the problem.

The development of sliding-mode controllers for

operation in the presence of single or multiple,

constant or time-varying state delays was solved by

Gouaisbaut, Dambrine, and Richard (2002). This uses

the equivalent control method where the exact knowl-

edge of block matrices must be known and the full-

state measurable. The work was further extended to

include polytopic uncertainties (Gouaisbaut, Blanco,

and Richard 2004), however full-state feedback was

required to yield finite reachability design and the

upper bound on the states spanning the input space is

assumed to be known for solution of the reachability

problem. The problem was also considered by Li and

DeCarlo (2003) where a class of uncertain time-delay

systems with multiple fixed delays in the system states

is considered. The method of equivalent control is used

and measurement of the state at the current time is

assumed. It is also assumed that the delayed state is

bounded with a bound dependent on the current state

which is restrictive. Even though an adaptive law was

used to estimate the bound of the uncertainty in the

reaching phase, a switching gain which is again

assumed to be large enough rather than calculated

explicitly is thought to guarantee the reachability

design. The work in Jafarov (2005) considers

*Corresponding author. Email: s.k.spurgeon@kent.ac.uk

ISSN 0020–7179 print/ISSN 1366–5820 online

� 2010 Taylor & Francis

DOI: 10.1080/00207179.2010.527375

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sliding-mode control of an uncertain system in thepresence of fixed state delay, but again full-statefeedback is assumed.

The assumption of full-state feedback is a limitingone in practice as it may be prohibitively expensive, andindeed, sometimes impossible, to measure all the statevariables. One approach to solve this problem is toimplement the controller with an observer, wherethe observer provides state estimates for use bythe controller. However, the implementation of thecontroller–observer is more involved and the theoreticalframeworks to ensure stability across the range ofpractical operation of the plant may be challenging.A more straightforward approach is to use only thesubset of state information that is available, i.e. themeasured output, within the control design paradigm.

There are typically two facets to the design of astatic output feedback sliding-mode control. One is theexistence problem, i.e. the design of a switching surfacein the output vector space which is usually of lowerorder than the state vector space. Consider first theswitching surface design problem for uncertain sys-tems, where time-delay effects are not involved. Twodifferent methods of designing sliding surfaces usingeigenvalue assignment and eigenvector techniques wereproposed in Zak and Hui (1993) and El-Khazali andDecarlo (1995). A canonical form via which the staticoutput feedback sliding-mode control design problemis converted to a static output feedback stabilisationproblem for a particular subsystem triple was providedin Edwards and Spurgeon (1995). However, the solu-tion to the general static output feedback problem,even for linear time-invariant systems, is still open.Iterative linear matrix inequality (LMI) approacheshave been exploited to solve the static output feedbackproblem using a bilinear matrix inequality formula-tion, see Cao, Lam, and Sun (1998), Choi (2002) andHuang and Nguang (2006). In Edwards (2004), wherethe regular form was not used for synthesis of thecontrol law, LMIs were derived for switching functiondesign whilst minimising the cost function associatedwith the control. Sufficient conditions for static outputfeedback controller design using LMIs have also beensought. Although only sufficient, the solutions havethe advantage of being linear and, hence, easilytractable using standard optimisation techniques, seeCrusius and Trofino (1999) and Shaked (2003). Thesecond facet in the design of a sliding-mode outputfeedback controller is the control, or reachability,synthesis problem whereby a control is determined toensure the sliding surface is attractive. It is non-trivialto synthesise a control law only using the outputvector, even for the situation where time-delay effectsare not considered, since the derivative of the slidingsurface is always related to the unmeasured states and

this derivative appears in the reachability condition. Aswell as within the existence problem, LMI methodshave also been considered within the context ofdeveloping a sliding-mode control strategy whichsolves the reachability problem for a given slidingsurface. For example, LMI methods which yieldreachability conditions for designing static sliding-mode output feedback controllers are presented inEdwards, Akoachere, and Spurgeon (2001).

In the context of output feedback sliding-modecontrol for time-delay systems, the existence andreachability problems for systems in the presence ofmatched uncertainty are considered in Han, Fridman,Spurgeon, and Edwards (2009). The delay is assumedto be time-varying and bounded where the upperbound is known. In line with the development ofoutput feedback controllers in the non-delayed case,LMIs are used to select all the parameters of theclosed-loop sliding-mode controller. However, noexplicit calculation of the switching gain in thenonlinear part of the control was given, it was onlyassumed to be large enough to induce the sliding mode.While asymptotic stability in the presence of matcheddisturbances can be achieved by sliding-mode control,unmatched disturbances usually lead to only boundedstability. For example, in Fernando and Fridman(2006) robustness properties of integral sliding-modecontrollers are studied where the Euclidean norm ofthe unmatched perturbation is minimised by selecting aprojection matrix.

The contribution of the proposed work is that allthe design parameters, including the switching gain, arederived from LMIs in spite of the presence of statedelay and unmatched disturbances. The method is ableto deal with polytopic type uncertainties in all blocksof the system matrices. No additional assumption ismade on the bound of the uncertain states in thereachability design, as required by other work. Somepreliminary results of this article are presented in Hanet al. (2009). Central to the work presented is thedescriptor approach (Fridman 2001), which is appliedto derive LMIs for the solution of the sliding-modeoutput feedback control problem in the presence ofboth matched and unmatched bounded disturbancesand time-varying state delays. It is demonstrated thatthe state trajectories of the system converge towards aball with a prespecified exponential convergence rate.In Section 2, the problem formulation is described andan appropriate general framework to accomplish theoutput feedback sliding-mode control design is formu-lated in Section 3. A constructive solution to theexistence problem is presented in Section 4 and Section5 shows the formulation of the reachability problemwhich will ensure that the sliding-mode is reached. Aproblem from the literature is used to provide a

2 X. Han et al.

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tutorial example of how the paradigm can be used tosolve both the existence and reachability problems forpractical design. A case study relating to the control ofan autonomous vehicle is used to further illustrate thedesign process in Section 6.

Notation: Rn denotes the n-dimensional Euclideanspace with vector norm k � k or the induced matrixnorm, Rn�m is the set of all n�m real matrices. P4 0,for P2Rn�n, denotes that P is symmetric and positivedefinite whereas * means the symmetric entries ofan LMI.

2. Problem formulation

Consider an uncertain dynamical system of the form

_xðtÞ ¼ AxðtÞ þ Adxðt� �ðtÞÞ þ BuðtÞ þ B1wðtÞ,

yðtÞ ¼ CxðtÞ,

xðt0 � �ðtÞÞ ¼ �ð�ðtÞÞ for �ðtÞ 2 ½0, h�,

ð1Þ

where x2Rn, u2Rm, w2Rk and y2Rp withm5 p5 n, � is absolutely continuous with squareintegrable _�, h is an upper bound on the time-delayfunction (0� �(t)� h 8t� 0). The time-varying delaymay be either slowly varying (i.e. a differentiabledelay with _�ðtÞ � d5 1) or fast varying (piecewise-continuous delay). Assume that the nominal linearsystem (A,Ad,B,B1,C ) is known and that the input andoutput matrices B and C are both of full rank. Thedisturbance is assumed to be bounded wherebykw(t)k�D with a known upper bound D4 0. A controlstrategy will be sought which induces an ideal slidingmotion with desirable performance characteristics onthe surface

S ¼ fx 2 Rn : sðtÞ ¼ FCxðtÞ ¼ 0g ð2Þ

for some selected matrix F2Rm�p so that the motion,when restricted to S, is stable.

3. A general framework for design

The first problem considered is how to choose F in(2) so that the associated sliding motion is stable. Acontrol law will then be determined to guarantee theexistence of a sliding motion. A convenient systemrepresentation closely allied to the usual regularform used for sliding-mode control design isemployed. It can be shown that if rank(CB)¼mand system triple (A,B,C ) are minimum phase, thereexists a coordinate system xr¼Trx, xr¼ [x1 x2]

T, in

which the system (A,Ad,B,B1,C ) has the trans-

formed structure

Ar ¼A11 A12

A21 A22

" #, Adr ¼

Ad11 Ad12

Ad21 Ad22

" #,

Br ¼0

B2

" #, B1r ¼

B11

B12

" #, Cr ¼ 0 T

� �, ð3Þ

where B22Rm�m is nonsingular and T2Rp�p is

orthogonal (Edwards and Spurgeon 1995).

Furthermore, A11, Ad112R(n�m)� (n�m) and the remain-

ing sub-blocks are partitioned accordingly. Let

F1 F2

� �¼ FT, ð4Þ

where F12Rm�( p�m) and F22R

m�m. As a result

FCr ¼ F1C1 F2

� �, ð5Þ

where

C1 ¼ 0ð p�mÞ�ðn�pÞ Ið p�mÞ� �

: ð6Þ

It is straightforward to see that FCrBr¼F2B2 and the

square matrix F2 is non-singular. By assumption, the

system contains both matched and unmatched uncer-

tainties and therefore the sliding motion is independent

of the matched uncertainty but dependent on the

unmatched uncertainty. In terms of the coordinate

framework defined above, the reduced-order sliding-

mode dynamics are governed by the following reduced-

order system

_x1ðtÞ ¼ ðA11 � A12KC1Þx1ðtÞ þ ðAd11 � Ad12KC1Þ

� x1ðt� �ðtÞÞ þ B11wðtÞ: ð7Þ

The response of this system must therefore be ulti-

mately bounded, where K ¼ F�12 F1, and the problem

of hyperplane design is equivalent to a static output

feedback problem for the system (A11,Ad11,A12,Ad12,

C1), where (A11þAd11,A12þAd12) is assumed control-

lable and (A11þAd11,C1) observable. Note that

the presence of the unmatched uncertainty means

that, in general, asymptotic stability cannot be

attained by the system (7). This is formalised in terms

of the existence problem, which must be solved to

determine the switching surface, in the following

section.

4. Existence problem

It will be shown that the system (3) is exponentially

attracted to a bounded region in Rn if the reduced-

order system (7) is exponentially attracted to a bounded

domain in Rn�m. Consider the Lyapunov–Krasovskii

International Journal of Control 3

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functional below for the exponential stability analysis

of (7)

VðtÞ ¼ xT1 ðtÞPx1ðtÞ þ

Z t

t�h

e�ðs�tÞxT1 ðsÞEx1ðsÞds

þ

Z t

t��ðtÞ

e�ðs�tÞxT1 ðsÞSx1ðsÞds

þ h

Z 0

�h

Z t

tþ�

e�ðs�tÞ _xT1 ðsÞR _x1ðsÞds d� ð8Þ

with (n�m)� (n�m)-matrices P4 0 and E� 0, S� 0,

R� 0. To prove exponential stability of the system (7)

using (8), we will use the following lemma.

Lemma 4.1 (Fridman and Dambrine 2009): Let V :

[0,1)!Rþ be an absolutely continuous function. If

there exist �4 0 and b4 0 such that the derivative of V

satisfies almost everywhere the inequality

d

dtVðtÞ þ �VðtÞ � bkwðtÞk2 � 0

then it follows that for all kw(t)k�D

VðtÞ � e��ðt�t0ÞVðt0Þ þb

�D2, t � t0:

Differentiating V(t) from (8) yields

M � 2xT1 ðtÞP _x1ðtÞ þ h2 _xT1 ðtÞR _x1ðtÞ

� he��hZ t

t�h

_xT1 ðsÞR _x1ðsÞdsþ xT1 ðtÞðEþ S Þx1ðtÞ

� xT1 ðt� hÞEx1ðt� hÞe��h þ �xT1 ðtÞPx1ðtÞ

� ð1� _�ðtÞÞxT1 ðt� �ðtÞÞSx1ðt� �ðtÞÞe���ðtÞ

� bwTðtÞwðtÞ: ð9Þ

Further using the identity

� h

Z t

t�h

_xT1 ðsÞR _x1ðsÞds

¼�h

Z t��ðtÞ

t�h

_xT1 ðsÞR _x1ðsÞds� h

Z t

t��ðtÞ

_xT1 ðsÞR _x1ðsÞds

ð10Þ

and applying Jensen’s inequalityZ t

t��ðtÞ

_xT1 ðsÞR _x1ðsÞds �1

h

Z t

t��ðtÞ

_xT1 ðsÞdsR

Z t

t��ðtÞ

_x1ðsÞds

ð11Þ

and Z t��ðtÞ

t�h

_xT1 ðsÞR _x1ðsÞds

�1

h

Z t��ðtÞ

t�h

_xT1 ðsÞdsR

Z t��ðtÞ

t�h

_x1ðsÞds ð12Þ

then Equation (9) becomes

M � 2xT1 ðtÞP _xT1 ðtÞ þ �xT1 ðtÞPx1ðtÞ þ h2 _xT1 ðtÞR _x1ðtÞ

�

h�x1ðtÞ � x1ðt� �ðtÞÞ

�TR�x1ðtÞ � x1ðt� �ðtÞÞ

���x1ðt� �ðtÞÞ � x1ðt� hÞ

�T� R

�x1ðt� �ðtÞÞ � x1ðt� hÞ

�ie��h

þ xT1 ðtÞðEþ S Þx1ðtÞ � xT1 ðt� hÞEx1ðt� hÞe��h

� ð1� d ÞxT1 ðt� �ðtÞÞSx1ðt� �ðtÞÞe��h

� bwTðtÞwðtÞ: ð13Þ

Using the descriptor method as in Fridman (2001) and

the free-weighting matrices technique from He, Wang,

Lin, and Wu (2007)

0� 2ðxT1 ðtÞPT2 þ _xT1 ðtÞP

T3 Þ½� _x1ðtÞþ ðA11�A12KC1Þx1ðtÞ

þ ðAd11�Ad12KC1Þx1ðt� �ðtÞÞþB11wðtÞ�, ð14Þ

where matrix parameters P2,P3¼ "P22Rn�m are

added to the right-hand side of (13). Setting �ðtÞ ¼colfx1ðtÞ, _x1ðtÞ, x1ðt� hÞ, x1ðt� �ðtÞÞ,wðtÞg, then M�

�T(t)��(t)� 0 if the matrix �5 0. Multiplying

matrix � from the right and the left by diagfP�12 ,

P�12 ,P�12 ,P�12 , I g and its transpose, respectively, and

denoting

Q2 ¼ P�12 , bP ¼ QT2PQ2, bR ¼ QT

2RQ2,bE ¼ QT2EQ2, bS ¼ QT

2SQ2

it follows �5 0, b�5 0 where

b� ¼b�11 b�12 0 b�14 b�15 b�22 0 b�24 b�25 b�33 b�34 0

b�44 0

b�55

26666664

377777755 0, ð15Þ

and

b�11 ¼ ðA11 � A12KC1ÞQ2 þ �bPþQT2 ðA11 � A12KC1Þ

T

þ bEþ bS� bRe��h,b�12 ¼ bP�Q2 þ �QT2 ðA11 � A12KC1Þ

T,b�14 ¼ ðAd11 � Ad12KC1ÞQ2 þ bRe��h,b�15 ¼ B11, b�22 ¼ �"Q2 � "QT2 þ h2bR,b�24 ¼ "ðAd11 � Ad12KC1ÞQ2, b�25 ¼ "B11,b�33 ¼ �ðbEþ bRÞe��h, b�34 ¼ bRe��h,b�44 ¼ �2e��hbR� ð1� d ÞbSe��h, b�55 ¼ �bI:

ð16Þ

4 X. Han et al.

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Select the LMI variable Q2 in the following form

Q2 ¼Q11 Q12

Q22M �Q22

� �, ð17Þ

where Q22 is a ( p�m)� ( p�m) matrix, M is a

( p�m)� (n� p) tuning matrix and � is a tuning

parameter to be selected by the designer. It follows that

KC1Q2 ¼ KQ22M �KQ22

� �:

Defining Y¼KQ22 it follows that

KC1Q2 ¼ YM �Y� �

: ð18Þ

To construct K, substitute (18) into (16) to yieldb�11 ¼ A11Q2 � A12½Y �Y � þQT2A

T11 þ �

bP,� ½YM �Y �TAT

12 þbEþ bS� bRe��h,b�12 ¼ bP�Q2 þ "Q

T2A

T11 � "½YM �Y �TAT

12,b�14 ¼ Ad11Q2 � Ad12½YM �Y � þ bRe��h,b�15 ¼ B11, b�22 ¼ �"Q2 � "QT2 þ h2bR,b�24 ¼ "Ad11Q2 � "Ad12½YM �Y �, b�25 ¼ "B11,b�33 ¼ �ðbEþ bRÞe��h, b�34 ¼ bRe��h,b�44 ¼ �2e��hbR� ð1� d ÞbSe��h, b�55 ¼ �bI:

ð19Þ

The following proposition can now be stated:

Proposition 4.2: Given scalars h4 0, d5 1, �4 0,

", �, b and a matrix M2R( p�m)�(n�p), if there exist

(n�m)� (n�m) matrices bP4 0, bE � 0, bS � 0, bR � 0

and matrices Q222R( p�m)�( p�m), Q112R

(n�p)�(n�p),

Q122R(n�p)�( p�m), Y2Rm�( p�m) such that the LMI

(15) with matrix entries (19) holds, then the reduced-

order system (7), where K ¼ YQ�122 , is exponentially

attracted to the ellipsoid

xT1 ðtÞPx1ðtÞ �b

�D2, ð20Þ

where P ¼ Q�T2bPQ�12 , for all differentiable delays

0 � �ðtÞ � h, _�ðtÞ � d5 1. Moreover, the reduced

order dynamics (7) is exponentially stable for all

piecewise-continuous delays 0� �(t)� h, if the LMI

(15) is feasible with bS ¼ 0.

Remark 1: Since the LMI (15) is affine in the system

matrices A, Ad and B1, the results are applicable to the

case where these matrices are uncertain. Denote �¼

[A Ad B1] and assume that �2Co{�j, j¼ 1, . . . ,N},

namely � ¼PN

j¼1 fj ðtÞ�j for some 0 � fj ðtÞ � 1,PNj¼1 fj ðtÞ ¼ 1, where the N vertices of the polytope

are described by �j ¼�Að j Þ A

ð j Þd B

ð j Þ1

�. One has to

solve the LMIs simultaneously for all the N vertices,

applying the same decision matrices for all vertices.

Example 4.3: Consider the following simple system

which is in regular form, with polytopic uncertaintiesand unknown (bounded) perturbations (t) and f(t)

_xðtÞ ¼

�3 2 1

2 1þ sinðx3ðtÞÞ 1

1 1 x22ðtÞ þ 1

264375xðtÞ

þ

0:5 0 0

0 1 0:2

�0:2 �0:5 1

264375xðt� �Þ

þ

0

0

1

264375 �uðtÞ þ

ðtÞx1ðtÞ þ 0:5wðtÞ

�0:5ðtÞx1ðt� �Þ � 0:5wðtÞ

0:2ðtÞx2ðtÞ þ wðtÞ

264375,

yðtÞ ¼0 1 0

0 0 1

� �xðtÞ ð21Þ

with 0�(t)� 2 and disturbance w(t)2 [�1, 1]. Thedelay is assumed to be time-varying. In order to

present (21) in the form of (1) with uncertain matrices,define the control variable �u(t) as follows:

�uðtÞ ¼ uðtÞ þ ðx22ðtÞ þ 1Þx3ðtÞ, ð22Þ

where u(t) is the sliding-mode control variable of theform (26). This change is possible because x2(t) andx3(t) are measured. Considering next sin(x3(t)) asuncertainty (t)¼ sin(x3(t))2 [�1, 1], the above systemis represented as a polytopic system with four verticesdefined by ¼1, ¼ 0 and ¼ 2

_xðtÞ ¼X4j¼1

fj ðtÞ Að j ÞxðtÞþA

ð j Þd xðt� �Þ

h iþB �uðtÞþB1wðtÞ,

ð23Þ

where

Að1Þ ¼

�3 2 1

2 2 1

1 1 0

264375, Að2Þ ¼

�3 2 1

2 0 1

1 1 0

264375,

Að3Þ ¼

�1 2 1

2 2 1

1 1:4 0

264375, Að4Þ ¼

�1 2 1

2 0 1

1 1:4 0

264375,

Að1Þd ¼ A

ð2Þd ¼

0:5 0 0

0 1 0:2

�0:2 �0:5 1

264375,

Að3Þd ¼ A

ð4Þd ¼

0:5 0 0

�1 1 0:2

�0:2 �0:5 1

264375,

B ¼

0

0

1

264375, B1 ¼

0:5

�0:5

1

264375: ð24Þ

International Journal of Control 5

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Note that state-feedback sliding-mode control of theabove system without unmatched disturbances andpolytopic uncertainties 0.2(t)x2(t) in _x3ðtÞ was con-sidered in Gouaisbaut et al. (2004). The work employsLMI methods for the solution of the existence prob-lem, and is suitable for uncertain systems where thepolytopic uncertainties appear only in the subsystem(7). The control law is derived based on the assumptionthat those states varying in the span of the controlinput are bounded so that a large enough switchinggain can induce the sliding motion in finite time. Anappropriate switching gain must usually be determinedby trial and error.

The advantage of the proposed method is that itfacilitates analysis of polytopic uncertainties appearingin the input channel and the switching gain is derivedfrom LMIs, ensuring finite time reachability onto thesliding surface with a prescribed decay rate.

The initial function is taken as x(t)¼ [1, 1�1]T fort2 [��, 0]. To construct K for the reduced-order system(7) according to Proposition 4.2, the parameter settingsin the LMI (15) with entries (19) are selected with thedelay upper bound h¼ 1 s and the rate of change ofthe time-varying delay _� � d ¼ 0:1. For �¼ 2, "¼ 0.3,M¼ 2 and choosing �¼ 0.1, b¼ 0.005, then it isobtained that the LMI variables

bP ¼ 949:4 39:5

925:9

" #, Y ¼ 1237:4,

Q22 ¼ 175, K ¼ 7:07:

Once a stable sliding-mode dynamics has beendesigned, the next step is to find a controller whichensures the closed-loop system reaches the prescribedsliding surface in finite time. This will now beconsidered in general terms.

5. Reachability problem

It can be shown (Edwards et al. 2001) that thefollowing system transformation and control structure

exist such that z(t)¼T1xr(t), where T1 ¼In�m 0KC1 Im

h iso

that the system ð �A, �Ad, �B,F �C Þ has the property

�A ¼�A11

�A12

�A21�A22

" #, �Ad ¼

�Ad11�Ad12

�Ad21�Ad22

" #,

�B ¼0

Im

" #, �B1 ¼

�B11

�B12

" #, F �C ¼ 0 Im

� �, ð25Þ

where z1(t)¼ x1(t), z2(t)¼ s(t), �A11¼A11�A12KC1 and�Ad11¼Ad11�Ad12KC1 exhibit the reduced-ordersliding-mode dynamics. Also, �C ¼ ½0 �T �, where�T 2 Rp�p is nonsingular. The control law is

considered as

uðtÞ ¼ �GyðtÞ � vyðtÞ, ð26Þ

where

G ¼ G1 G2

� ��T�1 ð27Þ

vyðtÞ ¼�

FyðtÞ

kFyðtÞk, if FyðtÞ 6¼ 0,

0, otherwise,

8<: ð28Þ

where G12Rm�( p�m), G22R

m�m, F¼ [K Im]T�1. The

uncertain system (1) becomes

_zðtÞ ¼ �AzðtÞ þ �Adzðt� �ðtÞÞ þ �BuðtÞ þ �B1wðtÞ: ð29Þ

Closing the loop in the system (29) with the control law

(26) yields

_zðtÞ ¼ A0zðtÞ þ �Adzðt� �ðtÞÞ � �BvyðtÞ þ �B1wðtÞ, ð30Þ

where A0 ¼ �A� �BG �C. Let �P be a symmetric positive

definite matrix partitioned conformably with (25)

so that �P ¼�P1 00 �P2

h i. It follows that �P �B ¼ ðF �C ÞTP2 and

from (25) Fy(t)¼ z2(t). It can be shown that

¼ �PA0 þ AT0

�P

¼

�P1�A11 þ �AT

11�P1

�P1�A12 þ ð �A21 � G1C1Þ

T �P2

�P2

�A22 þ �AT22

�P2

� �P2G2 � ð �P2G2ÞT

( )26643775

¼�P1

�A11 þ �AT11

�P1�P1

�A12 þ �AT21

�P2 � ðL1C1ÞT

�P2�A22 þ �AT

22�P2 � L2 � ðL2Þ

T

" #,

ð31Þ

where L1 ¼ �P2G1 and L2 ¼ �P2G2. A stability condition

for the full order closed-loop system can be derived

using the following Lyapunov–Krasovskii functional

VðtÞ ¼ zTðtÞ �PzðtÞ þ

Z t

t�h

e ��ðs�tÞzTðsÞ �EzðsÞds

þ

Z t

t��ðtÞ

e ��ðs�tÞzTðsÞ �SzðsÞds

þ h

Z 0

�h

Z t

tþ�

e ��ðs�tÞ _zTðsÞ �R _zðsÞds d�, ð32Þ

where �E� 0, �S � 0 and �R ¼�R1 00 0

h iwhere �R1 � 0 (as it is

desired to determine a stability condition for the time-

delay system which is delay independent of z2(t)). Then

�M ¼ _Vþ ��V� �bwTðtÞwðtÞ

� 2zTðtÞ �P _zTðtÞ þ ��zTðtÞ �PzðtÞ þ h2 _zTðtÞ �R _zðtÞ

�

hðzðtÞ � zðt� �ðtÞÞÞT �RðzðtÞ � zðt� �ðtÞÞÞ

6 X. Han et al.

Downloaded By: [Steklyar, Inna] At: 07:06 15 November 2010

þ ðzðt� �ðtÞÞ � zðt� hÞÞT

� �Rðzðt� �ðtÞÞ � zðt� hÞÞie� ��h

þ zTðtÞð �Eþ �S ÞzðtÞ � zTðt� hÞ �Ezðt� hÞe� ��h

� ð1� d ÞzTðt� �ðtÞÞ �Szðt� �ðtÞÞe� ���ðtÞ � �bwTðtÞwðtÞ:

ð33Þ

Substitute the right-hand side of Equation (30) into (33).

Setting &(t)¼ col{z(t), z(t� h), z(t� �(t)),w(t)}, then

_VðtÞ � &ðtÞT�h&ðtÞ þ h2 _zTðtÞ �R _zðtÞ

þ 2zT �P �Bð �B12wðtÞ � vyðtÞÞ5 0 ð34Þ

is satisfied if &TðtÞ�h&ðtÞ þ h2 _zTðtÞ �R _zðtÞ5 0 and

2zT �P �Bð �B12wðtÞ � vyðtÞÞ5 0, where

�h ¼

�11 0 �P �Adþ �Re� ��h�P1

�B11

0

" # �22 �Re� ��h 0

�2e� ��h �R�ð1� d Þ �Se� ��h 0

� �bI

266666664

377777775ð35Þ

with

�11 ¼ þ �� �Pþ �Sþ �E� �Re� ��h; �22 ¼ �ð �Eþ �RÞe� ��h:

Setting �(t)¼ col{z(t), z(t� h), z(t� �(t)),w(t), vy(t)}

and �I ¼ Iðn�mÞ 0� �T

, it is obtained that

h2 _zTðtÞ �R _zðtÞ

¼ zTðtÞAT0 þ zTðt� �ðtÞÞ �AT

d � vTy ðtÞ�BT þ wTðtÞ �BT

1

h i� h2 �R A0zðtÞ þ �Adzðt� �ðtÞÞ � �BvyðtÞ þ �B1wðtÞ

� �

¼ �TðtÞ

AT0

0

�ATd

�BT1

�BT

26666664

37777775 �Ih2 �R1�IT

AT0

0

�ATd

�BT1

�BT

26666664

37777775

T

�ðtÞ: ð36Þ

Using the Schur complement, �TðtÞ�h�ðtÞ þh2 _zTðtÞ �R _zðtÞ5 0 holds if

�h

hAT0

Iðn�mÞ

0

� ��R1

0

h �ATd

Iðn�mÞ

0

� ��R1

h �BT1

Iðn�mÞ

0

� ��R1

� �R1

26666666666664

377777777777755 0 ð37Þ

for some ��4 0, �b4 0 and 0� �(t)� h, i.e. to ensure the

exponential attractiveness of (30) to the ellipsoid

zTðtÞ �PzðtÞ ��b��D

2. Given the control structure in (27),

then

2zTðtÞ �P �Bð �B12wðtÞ � vyðtÞÞ

¼ 2zT2 ðtÞ�P2ð �B12wðtÞ � vyðtÞÞ

� �2� �P2kz2ðtÞk þ 2 �P2k �B12kkz2ðtÞkD

5 0:

The latter inequality implies exponential attractivity

of the ellipsoid zTðtÞ �PzðtÞ ��b��D

2, thus for t!1,

zTðt� �ðtÞÞ �Pzðt� �ðtÞÞ ��b��D

2 holds. The following

proposition can now be stated:

Proposition 5.1: Given scalars h4 0, d5 1, ��4 0,�b4 0, assume there exist n� n matrices �P ¼ diagf �P1,�P2g4 0 with P22R

m�m, �E � 0, �S � 0, a (n�m)�

(n�m)-matrix �R1 � 0, L12Rm�( p�m), L22R

m�m such

that LMI (37) is feasible. Then for �4 k �B12kD the

closed-loop system (30), where G1 ¼ �P�12 L1, G2 ¼�P�12 L2, is exponentially attracted to the ellipsoid

zTðtÞ �PzðtÞ ��b��D

2 for all �(t)2 [0, h]. Consequently it

also holds that zTðt� �ðtÞÞ �Pzðt� �ðtÞÞ ��b��D

2 for t!1.

Denote

AL0 ¼ ½0 Im�A0, �AL

d ¼ ½0 Im� �Ad, ¼�b

��D2: ð38Þ

Given �14 0, �24 0, conditions will now be derived

that guarantee the solutions of (25) satisfy the bound

kAL0 zðtÞk5 �1, kA

Ld zðt� �ðtÞÞk5 �2 ð39Þ

for t!1. The following inequalities

zTðtÞðAL0 Þ

TðAL

0 ÞzðtÞ

� �21zTðtÞ �PzðtÞ

zTðt� �ðtÞÞðALd Þ

TðAL

d Þzðt� �ðtÞÞ

� �22zTðt� �ðtÞÞ �Pzðt� �ðtÞÞ

ð40Þ

guarantee (39). Hence equivalently

ðAL0 Þ

TðAL

0 Þ ��21

�P

, ðAL

d ÞTðAL

d Þ ��22

�P

ð41Þ

or by Schur complements

��21

�P

ðAL

0 ÞT

�I

24 355 0,��22

�P

ðAL

d ÞT

�I

24 355 0:

ð42Þ

International Journal of Control 7

Downloaded By: [Steklyar, Inna] At: 07:06 15 November 2010

Theorem 5.2: Given scalars ��4 0, �b4 0, let thereexist n� n matrices �P ¼ diagf �P1, �P2g4 0, �E � 0,�S � 0, a (n�m)� (n�m)-matrix �R1 � 0, L12

Rm�( p�m), L22R

m�m such that the LMI (37) is feasiblefor 0� �(t)� h, _�ðtÞ � d5 1. Let �1 and �2 satisfy (42)with the notation given in (38). Then for

�4 kB12kDþ �1 þ �2 ð43Þ

an ideal sliding motion takes place on the surface S. Theclosed-loop system (26), (29) is ultimately bounded by

lim supt!1 zTðtÞ �PzðtÞ ��b

��D2:

Proof: Substituting the control law it follows from(29) that

_sðtÞ ¼ F �CA0zðtÞ þ F �C �Adzððt� �ðtÞÞ þ ð �B12wðtÞ � vyðtÞÞ:

Let Vc :Rm!R be defined by VcðsÞ ¼ sTðtÞ �P2sðtÞ. It

follows that

�P2F �CA0 ¼ �P2AL0 ,

�P2F �CAd ¼ �P2ALd :

Starting from initial condition z(t0), it can be verifiedthat there exists t14 0 such that for all t� t1,

_VcðsÞ ¼ 2sTðtÞ �P2AL0 zðtÞ þ 2sTðtÞ �P2A

Ld zðt� �ðtÞÞ

þ 2sTðtÞ �P2ð �B12wðtÞ � vyðtÞÞ

� 2ksðtÞkk �P2kðkAL0 zðtÞk þ kA

Ld zðt� �ðtÞÞkÞ

� 2ð�1 þ �2ÞksðtÞkk �P2k

5 �2�ksðtÞk, ð44Þ

where � ¼ �1 þ �2 � kAL0 zðtÞk � kA

Ld zðt� �ðtÞÞk. A slid-

ing motion will thus be attained in finite time. œ

Remark 2: Since LMIs (15), (37) and (42) are affine inthe system matrices A, Ad and B1, the results are

applicable to the case where these matrices are uncer-

tain with polytopic type uncertainties (see Remark 1).

One has to solve the LMIs simultaneously for all the N

vertices, applying the same decision matrices for all

vertices. In contrast to the existing methods in the

literature (Gouaisbaut et al. 2004; Seuret, Edwards,

Spurgeon, and Fridman 2009), polytopic type uncer-

tainties can be incorporated in all the blocks ofA,Ad,B1

and not only in A11, Ad11 because the switching gain �(and not only the sliding surface) is found using LMIs.

Example 5.3: Following on from Example 4.3, where

a sliding surface prescribing stable dynamics has been

designed for the uncertain system (21), then the control

law in (27) will have the sliding function matrix

F¼ [7.07, 1]. A control gain G must be designed

which will bring the closed-loop system into a bounded

region centred at the sliding surface. Setting �� ¼ 0:3,�b ¼ 5 in Proposition 5.1, it is obtained that

�P¼

22:4 �12:8 0

29 0

0:68

26643775, L1 ¼�6:08, L2 ¼ 85:4,

which gives

G ¼ �9, 126:6� �

�T�1, where �T ¼1 0

�7:07 1

� �:

Once the state of the closed-loop system has entered

the sliding patch zTðtÞ �PzðtÞ ��b��D

2, the switching gain

�¼ 753 derived from LMI (42) will ensure the sliding

surface is reached in finite time. Figure 1 shows that

the sliding surface is reached in finite time and the

outputs of the system are stable with ultimate bound

ky(t)k� 0.2.Sliding-mode control has the advantage over

linear control for its absolute rejection of the

0 1 2 3 4 5 6 7 8 9 10−7

−6

−5

−4

−3

−2

−1

0

1

2

Time (s)

Out

puts

4 6 8 10−0.4

−0.2

0

0.2

0.4

0 2 4 6 8 10−5

0

5

10

15

20

25(b)(a)

Time

Slid

ing

surf

ace

Figure 1. Closed-loop response with delay h¼ 1 s: (a) outputs and (b) sliding surface.

8 X. Han et al.

Downloaded By: [Steklyar, Inna] At: 07:06 15 November 2010

matched uncertainties. To verify the statement, sup-

pose there is a change of the matched disturbance at

time 10 s of magnitude from 1! 50, then the sliding

surface remains unaffected so that B1¼ [0 0 50]T. While

keeping the same control parameters obtained so far

comparisons between using sliding-mode control and

only the linear control G for the new closed-loop

design with only matched disturbances are made in

Figure 3. As can be seen, the sliding-mode control is

more robust than the linear control to matcheddisturbances. The difference between using the linearpart of the control G alone and SMC with switchingfor a system with unmatched disturbances can bedemonstrated below. For the same uncertain system,suppose the unmatched disturbances are changed fromB1¼ [0.5,�0.5, 1]T to B1¼ [2,�2, 1]T after the initial10 s. Using the linear control G¼ [25, 4] alone in thefeedback, the responses for the outputs y(t), slidingfunction s(t) and control input u(t) are plotted inFigure 2(a). As can be seen the outputs ky(t)k� 1.4,sliding function is bounded as ks(t)k� 1.16 and controlinput ku(t)k� 4.7. If SMC is used with the same lineargain and a switching gain �¼ 7, the system responsesare plotted in Figure 2(b), where ky(t)k� 0.81, s(t)¼ 0,ku(t)k� 7. Therefore for a system with unmatcheddisturbances, SMC can give an ideal sliding surfaceresponse rather than the bounded sliding functiongiven by its linear control part. As a result, a smallerbound of the outputs can be obtained. For linearcontrol to yield the similar bound on the outputs as bySMC, the linear gain needs to be increased fromG¼ [25, 4] to G¼ [134, 21] as seen in Figure 2(c), whereky(t)k� 0.83. To conclude, for a linear control to givesimilar output performance in the presence ofunmatched disturbances, the linear gain needs to belarger, but not substantially larger than the linear partof the SMC. In another words, a linear control designfor a system with unmatched disturbances can give asimilar output bound as SMC if the linear control islarge enough.

6. Autonomous vehicle control

Autonomous vehicles are expected to operate effec-tively in time-varying and uncertain conditions. Here acase study described in Yao, Spurgeon, and Edwards(2006) is considered, where the elevation angle of thegun barrel of a vehicle in space should be maintainedwhile the hull of the vehicle is subject to externaldisturbances resulting from the motion of the vehicleacross rough terrain. To meet the high controlspecifications, sliding-mode control has been consid-ered within the application domain for its robustnessagainst friction and disturbances and its ease ofimplementation for motor drive control. The existenceproblem must determine a sliding surface that mini-mises the ultimate bound of the reduced-order dynam-ics in the presence of time-varying state delay andunmatched disturbances relating to frictional effects.A fully nonlinear simulation model of the system isavailable for controller analysis and testing (Yao et al.2006). The model is physically based and is known to

0 2 4 6 8 10 12 14 16 18 20−2

0

2(a)

(b)

(c)

X : 17.5Y : 1.393

Out

puts

0 2 4 6 8 10 12 14 16 18 20−2

0

2

X : 18.08Y : 1.16

Slid

ing

func

tion

0 2 4 6 8 10 12 14 16 18 20

−5

0

5

X : 18.17Y : −4.683

Inpu

t

0 2 4 6 8 10 12 14 16 18 20−1

0

1X : 17.91Y : 0.8051

Out

puts

0 2 4 6 8 10 12 14 16 18 20−10

0

10

Slid

ing

surf

ace

0 2 4 6 8 10 12 14 16 18 20−50

0

50

X : 17.42Y : 7

Con

trol

inpu

t

0 2 4 6 8 10 12 14 16 18 20−1

0

1X : 17.7Y : 0.8334

Out

puts

0 2 4 6 8 10 12 14 16 18 20−0.5

0

0.5

X : 16.74Y : −0.1176

Slid

ing

func

tion

0 2 4 6 8 10 12 14 16 18 20−5

0

5

X : 16.66Y : 2.642

Time

Inpu

t

Figure 2. Closed-loop response with linear control andSMC: (a) linear control with G, (b) SMC with linear part Gand (c) linear control with G.

International Journal of Control 9

Downloaded By: [Steklyar, Inna] At: 07:06 15 November 2010

represent with high fidelity the dynamics and behav-

iour of the real system:

_xðtÞ ¼

�Dm

JmN2�

Km

JmN

Dm

JmN0 0

1

N0 �1 0 0

Dm

J1N

Km

J1

�Dm�D12

J1

�K12

J1

D12

J10 0 1 0 �1

0 0D12

J2

K12

J2

�D12

J2

26666666666664

37777777777775|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

A

xðtÞ

þ

Kt

Jm0

0

0

0

266666664

377777775|fflfflffl{zfflfflffl}

B

uðtÞ

þ

�DmðN� 1Þ

JmN2�

1

Jm0

1

JmN20

N� 1

N0 0 0 0

DmðN� 1Þ

J1N0 �

1

J10

1

J10 0 0 0 0

0 0 0 0 0

2666666666664

3777777777775|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

B11

wðtÞ,

ð45Þ

where the state vector xðtÞ ¼ _�mðtÞ, �mbðtÞ, _�bðtÞ,�

�blðtÞ, _�l ðtÞ�T, and wðtÞ ¼ _�pðtÞ,!1mðtÞ,!1lðtÞ,

�Fmb�

signð _�mðtÞ � _�pðtÞÞ, Fmb signð _�bðtÞ � _�pðtÞÞ�T:The friction

related signals are

FmbðtÞ ¼ K�Dm

N_�mðtÞ þ

DmðN� 1Þ

N_�pðtÞ þ Km�mbðtÞ

,!1mðtÞ ¼ fms � signð�amðtÞ � Jm €�pðtÞÞ,

!1lðtÞ ¼ fls � signð�a1ðtÞ � J1 €�pðtÞÞ,

where the second and third states of the disturbance

from w(t) are a function of the friction level fd, where

fd can take values 0, 1, 2, 3 (Yao et al. 2006). Note

�mb ¼ 1N �m � �b þ ð1�

1N�p;

_�b breech velocity;�m motor position;_�l muzzle velocity;_�p pitch rate disturbance;Jm motor inertia;N gearbox of ratio;J1 elevation inertia on load one;

Km and Dm stiffness and damping

between the motor and the

load;K12 and D12 stiffness and damping

between the load one and

the load two;�am and �al applied torque to the motor

and the load;!1m and !1l motor friction and the load

friction;

0 1 2 3 4 5 6 7 8 9 10−7

−6

−5

−4

−3

−2

−1

0

1

2

Time (s)

Out

puts

2 4 6 8 10−0.2

−0.1

0

0.1

0.2

0.3

Sliding-mode control (y1)

Sliding-mode control (y2)

Linear control (y1)

Linear control (y2)

Figure 3. Comparison between sliding mode control and linear control in the presence of matched disturbance and delay.Available in colour online.

10 X. Han et al.

Downloaded By: [Steklyar, Inna] At: 07:06 15 November 2010

fms and fls motor coulomb friction andthe load coulomb friction;

u control input defined as thevoltage input to the poweramplifier;

_�p disturbance input defined aspitch rate disturbance;

!1m and signð _�m � _�pÞ motor friction disturbances;

!1l and signð _�b � _�pÞ load friction disturbances

The parameter values used in Yao et al. (2006)define

A¼

�338:14 �2:55� 107 50942 0 0

0:0066 0 �1 0 0

0:66 50000 �110:1 �15000 10

0 0 1 0 �1

0 0 7:69 11538 �7:69

26666664

37777775,

B¼

4523:1

0

0

0

0

26666664

37777775,

B11 ¼

�50604 �769 0 5:1 0

1 0 0 0 0

99 0 �0:01 0 0:01

0 0 0 0 0

0 0 0 0 0

26666664

37777775,

C¼

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

264375, ð46Þ

and the disturbance is known to be

k _�pk � 0:07, k!1mk � 0:5� fd, k!1lk � 10� fd,

kFmb signð _�m � _�pÞk � 4, kFmb signð _�b � _�pÞk � 4:

ð47Þ

The vehicle dynamics is augmented with an extra staterelated to the breech position, where the desired breechposition is zero. Denote Ca¼ [0 0 1 0 0], the state spacerepresentation of the augmented system is given by

Ac ¼0 �Ca

0 A

� �, Bc ¼

0

B

� �, Cc ¼

1 0

0 C

� �:

ð48Þ

6.1 Sliding-mode control design

A design which does not incorporate knowledge ofdelay effects is first performed to yield a benchmarklevel of performance. Firstly it is necessary to construct

K for the reduced-order system (7) according toProposition 4.2. The parameter settings in LMI (15)with entries (19) are selected as Ad¼ 0, h¼ 0 s. If

� ¼ 11:5, " ¼ 0:0000002, M ¼

0:15 0:015

0:21 0:06

0:009 0:0003

264375

and choosing �¼ 7.2, b¼ 0.0005, then it is obtainedthat

K ¼ 0:9, 28, 327� �

: ð49Þ

The poles of the corresponding reduced order systemare

�4:15 j106, �389:6 j160:7, �2823:7� �

: ð50Þ

The control law in (27) will have the slidingfunction matrix

F ¼ �327 0:0002 28 0:9� �

: ð51Þ

A control G is designed which will bring the closed-loop system into a bounded region centred about thesliding surface. Setting �� ¼ 0:8, �b ¼ 3:88� 10�6 inProposition 5.1, it is obtained that

G ¼ �1:02� 107, 7:6, 910610, 27834� �

: ð52Þ

The closed-loop poles of Ac�BcGCc are

�31257 �2810:4 �390:8 j160:9 �4:2 j106� �

:

The switching gain �¼ 1561, which is derived fromLMI (42), will ensure the sliding surface is reached infinite time. Figure 4 shows the position error and rateerror for fd¼ 1, 2, 3 using the proposed controller. Inthe original case study (Yao et al. 2006), an observerwas used to estimate the effect of the disturbance andthe equivalent control method was used to synthesisethe control law, which was augmented with an addi-tional PI control. With this strategy the position andrate errors were kep(t)k� 0.2� 10�3 rad and ker(t)k�0.01 rad/s respectively. Setting a fixed sampling fre-quency of 10 kHz and choosing ode3 solver insimulink, kep(t)k� 0.8� 10�5 rad, ker(t)k� 0.4�10�3 rad/s was achieved, as seen in Figure 4, forfd¼ 1, 2, 3 with the proposed control scheme. Theoutput feedback sliding-mode control approach pre-sented in this section has thus improved the trackingaccuracy over previous results in Yao et al. (2006).The ultimate bound of the outputs is a function of theunmatched disturbance, but it can be seen that theeffect of the friction disturbance on the controlperformance after changing fd¼ 1,! 3 is very small.

Speed control of a motor in the presence ofuncertainties such as friction normally exhibit delaysdue to the fact that the mechanical response of the

International Journal of Control 11

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motor is slower than the electrical command. The sizeof the delay will depend upon the physical parametersof the actuator and can vary from milliseconds toseveral seconds, depending on the application. To takeaccount of such delay effects in the actual system forthe control design purpose, it is desirable to introducea system model incorporating delay into the systemmodel used for design. This will provide a means toanalyse the potential delay effect on the stability of theclosed-loop system at the design stage. Assume thedelay matrix

Ad ¼

�10 20 0 0 0

0:007 0 0:1 0 0

0 20 �2 1 0

0 0 0:1 0 0:1

0 0 2 1 0

26666664

37777775and the augmented matrix Adc ¼

0 0

0 Ad

� �:

ð53Þ

The open-loop tests on system (46) with the delaymatrix Ad in (53) shows that the vehicle system withstate delay h� 3ms yields a breech position error ofkep(t)k� 0.01 rad/s as expected from the known systemresponse. The augmented linear system with delay hasdynamics close to those of the original plant.

Designing a controller without considering explic-itly possible delay effects within the control designprocess can lead to deterioration of the system perfor-mance and sometimes even instability. Suppose there isa constant delay h¼ 3ms in the system where the valuesof F and G are taken as in (51) and (52) respectively,with the delay distribution matrix in (53). In this casethe position error will increase from kep(t)k� 0.8� 10�5

to kep(t)k� 1.4� 10�3 rad. The closed-loop systembecomes unstable for h� 4ms when delay effects arenot incorporated in the design process.

A controller will now be designed based on a modelincorporating delay effects. Firstly to construct Kfor the reduced-order system (7) according toProposition 4.2, the parameter settings in LMI (15)with entries (19) are selected with the delay upperbound h¼ 10ms and the rate of change of the time-varying delay _� � d ¼ 0. If for

� ¼ 3, " ¼ 0:0013, M ¼

0:0018 0:002

0:22 0:22

7:54 1:1

264375

and choosing �¼ 0.9, b¼ 0.0048, then it is obtainedthat

K ¼ 0:01, 16, 16:8� �

: ð54Þ

The poles of the reduced-order system are

�5:23 j92:9, �71:3 j254:8, �477:8� �

: ð55Þ

Thus the control law in (27) will have the slidingfunction matrix

F ¼ �16:8, 0:0002, 16, 0:01� �

:

The control G is designed to bring the closed-loopsystem into a bounded region centred about thesliding surface. Setting �� ¼ 0:8, �b ¼ 3:88� 10�6 inProposition 5.1, it is obtained that

G ¼ �2:02� 106, 26:7, 1:94, 0:002� �

:

The closed-loop poles of Ac�BcGCc are

�120700, �481:1, �75:2 j253:2, �5:43 j92:9� �

:

The switching gain �¼ 68,632, which is derived fromLMI (42), will ensure the sliding surface is reachedin finite time. The initial function was chosen asx(t0� �)¼ 0 for � 2 [0 h] in the simulation. The closed-loop performance is shown in Figure 5 for fd¼ 1, 2, 3.The position and rate errors are kept within the bound

0 1 2 3 4 5−1

−0.8−0.6−0.4−0.2

00.20.40.60.8

1 x 10−5

Time (s)

Pos

ition

err

or (

rad)

fd=1fd=2fd=3

0 1 2 3 4 5−4

−3

−2

−1

0

1

2

3

4(b)(a)x 10−4

Time (s)

Rat

e er

ror

(rad

/s)

fd=1fd=2fd=3

Figure 4. Closed-loop response without delay: (a) position error and (b) rate error. Available in colour online.

12 X. Han et al.

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kep(t)k� 0.8� 10�4 rad, ker(t)k� 0.1� 10�4 rad/s in thepresence of delay. Despite the effect of the frictiondisturbance on the nonlinear model which is not fullyrejected, the controller is seen to be robust to thedisturbance even in the presence of delay. This hasdemonstrated the efficiency of the proposed controlscheme on a system of practical interest.

7. Conclusion

The development of output feedback-based sliding-mode control schemes for systems in the presence ofstate delay and both matched and unmatched distur-bances has been presented. A descriptor Lyapunovfunctional approach has been used for switchingfunction design. The methodology has been imple-mented using LMIs and can give desirable sliding-mode dynamics. The advantage of the method is thatfor the first time and despite only output feedbackbeing available, not only the switching function isderived from LMIs but also the switching gainrequired to solve the reachability problem is deter-mined using LMIs. The method allows polytopicuncertainties to be included in all blocks of A, Ad, B1

and not only in A11, Ad11 as with other methods. Thisis novel even for systems without delay. As well as anexample incorporating polytopic uncertainties, themethodology has also been applied to a nonlinearautonomous vehicle control problem. Nonlinear sim-ulations show that the gun barrel is maintained at thedesired position, despite variation in the vehicle motioncaused by friction.

Acknowledgements

The authors gratefully acknowledge EPSRC support viaresearch grant EP/E 02076311.

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